Bayesian search for gravitational wave bursts in pulsar timing array data Bence B´ecsy, Neil J. Cornish eXtreme Gravity Institute, Department of Physics, Montana State University, Bozeman, Montana 59717, USA E-mail: [email protected] April 2021 Abstract. The nanohertz frequency band explored by pulsar timing arrays provides a unique discovery space for gravitational wave signals. In addition to signals from anticipated sources, such as those from supermassive black hole binaries, some previously unimagined sources may emit transient gravitational waves (a.k.a. bursts) with unknown morphology. Unmodeled transients are not currently searched for in this frequency band, and they require different techniques from those currently employed. Possible sources of such gravitational wave bursts in the nanohertz regime are parabolic encounters of supermassive black holes, cosmic string cusps and kinks, or other, as-yet-unknown phenomena. In this paper we present BayesHopperBurst, a Bayesian search algorithm capable of identifying generic gravitational wave bursts by modeling both coherent and incoherent transients as a sum of Morlet-Gabor wavelets. A trans-dimensional Reversible Jump Markov Chain Monte Carlo sampler is used to select the number of wavelets best describing the data. We test BayesHopperBurst on various simulated datasets including different combinations of signals and noise transients. Its capability to run on real data is demonstrated by analyzing data of the pulsar B1855+09 from the NANOGrav 9-year dataset. Based on a simulated dataset resembling the NANOGrav 12.5-year data release, we predict that at our most sensitive time-frequency location we will be able to probe gravitational wave bursts with a root-sum-squared amplitude higher than ∼ 5 × 10−11 Hz−1=2, which corresponds to 2 arXiv:2011.01942v2 [gr-qc] 14 Apr 2021 ∼ 40M c emitted in GWs at a fiducial distance of 100 Mpc. 1. Introduction The detection of nanohertz gravitational waves (GWs) is the main objective of pulsar timing arrays (PTAs). These large scale experiments regularly monitor a collection of millisecond pulsars to achieve this goal (see e.g. [1]). The three major pulsar timing arrays currently operating are the North American Nanohertz Observatory for Gravitational Waves (NANOGrav, [2]), the European Pulsar Timing Array (EPTA, [3]), and the Parkes Pulsar Timing Array (PPTA, [4]). In addition, there are emerging PTA efforts in India (InPTA, [5]), China (CPTA, [6]), and South Africa (SAPTA, [7]). These Bayesian search for gravitational wave bursts in PTA data 2 project are collaborating under the International Pulsar Timing Array (IPTA, [8, 9, 10]) consortium. The most promising GW sources in the nHz regime are supermassive black hole binaries (SMBHBs). These can be detected through observing their collective effect as a red noise process with quadrupolar correlations between pulsars (see e.g. [11]). The most massive and nearby of these SMBHBs might also be detectable individually (see e.g. [12]). It has also been proposed that these two searches can be carried out simultaneously, thus accounting for any interaction between the two types of signatures [13]. Searches have also been carried out for bursts with memory [14], i.e., a permanent deformation of spacetime after a violent astrophysical event, like the merger of two black holes. In this paper, we focus on searching for generic GW transients (a.k.a. GW bursts) in PTA data. The subject has been studied extensively in the context of ground-based interferometric GW detectors, where many different algorithms are in use [15, 16, 17, 18, 19, 20] and several searches have been carried out throughout different observing runs of the LIGO [21] and Virgo [22] detectors [23, 24]. The detection problem of GW bursts has also been considered for space-based detectors [25]. In the PTA context, no analysis of real data has been carried out so far. However, several methods have been suggested, including an analytical Bayesian framework [26], a Bayesian nonparametric approach [27], and a frequentist search working in the time- frequency domain [28]. Ref. [29] suggests a possible improvement by considering the coherence between the pulsar terms, which appears in some cases. In this paper we present BayesHopperBurstz, a Bayesian search for GW bursts in PTA data, which is based on a trans-dimensional Reversible Jump Markov Chain Monte Carlo (RJMCMC) sampler [30, 31] akin to that used in the BayesWave algorithm [15] to search for and reconstruct GW bursts in the data of ground-based GW detectors. The work described in this paper is the continuation of that presented in [32], where the authors used similar methods to model noise transients in PTA data. One promising source of nHz GW bursts are parabolic (or highly eccentric) encounters of two SMBHs, which can occur in a hierarchical triple system of SMBHs [33]. Cosmic string kinks and cusps can also produce GW busts in this frequency range [34]. The paper is organized as follows. In Section 2 we describe the methods employed by BayesHopperBurst including the model used to describe transient signals and the sampling techniques that enables it to effectively explore the parameter space. In Section 3 we perform various injection tests, demonstrating that BayesHopperBurst recovers a wide variety of signals and noise transients. In Section 4 we analyze a simulated dataset similar to the latest NANOGrav data release [2] to make a prediction of what we can expect from analyzing the real dataset. In Section 5 we analyze B1855+09 from the NANOGrav 9-year dataset [35] to demonstrate our algorithm's noise transient modeling capabilities on real data. Finally, we offer concluding remarks z https://github.com/bencebecsy/BayesHopperBurst Bayesian search for gravitational wave bursts in PTA data 3 in Section 6. 2. Methods In this section, we introduce the model used to describe signals and noise transients with arbitrary morphology, and we provide some details of the sampling techniques employed to efficiently explore the resulting high-dimensional parameter space. We use the enterprisex [36] software package for handling PTA data and calculating the likelihood used in our Bayesian analysis. Timing models were produced by libstempok, which is a python wrapper for the tempo2{ timing package. We also make use of the la forge+ [37] package for some of our figures. 2.1. Model The model we use to describe our dataset has many similarities with the one used in the BayesHopper algorithm. The only fundamental difference is that the collection of sinusoids are replaced by a set of \signal" wavelets coherent across detectors and a set of \noise transient" wavelets which only appear in a given pulsar and describe transient noise features in the data. In this section we give a brief description of the model and focus on differences between BayesHopper and BayesHopperBurst. More details can be found in [13]. Consider a PTA consisting of N pulsars. The ith residual in the kth pulsar of the array is modeled as: X δtki = Mkilδξkl + nki + gki + wki(θs) + vki(θn); (1) l where δξkl is the offset from the best fit value of the lth timing model parameter, and Mkil is the design matrix, which represents a timing model linearized around the best fit parameter values. nki represents all noise processes unique to each pulsar, while gki is the contribution of an isotropic stochastic GW background (GWB) which is correlated between different pulsars. The contribution of the transient GW signal is represented by wki(θs), while that of incoherent transient noise features is described by vki(θn), where θs and θn represent the parameters describing the GW signal and the transient noise respectively. 2.1.1. Transient noise model The contribution of noise transients to the timing residual (vki(θn)) is modeled as a sum of sine-Gaussian (Morlet-Gabor) wavelets: M Xk vki(θn) = Ψ(tki; λkj); (2) j=1 x https://github.com/nanograv/enterprise k https://github.com/vallis/libstempo { https://bitbucket.org/psrsoft/tempo2/ + https://github.com/Hazboun6/la_forge Bayesian search for gravitational wave bursts in PTA data 4 where Mk is the number of wavelets used in the kth pulsar, tki is the ith observing time of the kth pulsar, and λkj is the parameter vector describing the jth wavelet in the kth pulsar, which is related to the full parameter vector as θn = (M1; λ11; :::; λ1M1 ; :::; MN ; λN1; :::; λNMN ), and the wavelets are defined as: 2 2 (t−t0) /τ Ψ(ti; λkj) = Ae cos(2πf0(t − t0) + φ0); (3) where A is the amplitude, t0 is the central time, τ is the characteristic duration, f0 is the central time, and φ0 is the initial phase. We can see that a single wavelet can be described by 5 parameters, i.e. λkj = (A; t0; τ; f0; φ0). We use Morlet-Gabor wavelets as they have the compelling property of having the smallest time-frequency area allowed by the Heisenberg{Gabor limit [38]. Other functions, such as shapelets [39], might be better suited to certain types of signals, so we plan to investigate using them in the future. 2.1.2. Signal model In order to impose the proper coherence between pulsars, in the signal model we model the GW waveform itself as a sum of wavelets, and then project that onto the pulsar lines of sight taking into account the corresponding antenna factors. Since PTAs are sensitive to the time integral of the metric perturbation, we introduce: Z H+;×(t) = h+;×(t)dt; (4) where h+;× are the usual components of the metric perturbation corresponding to plus and cross polarized GWs. We model H+;×(t) as: N X H+(t) = Ψ(t; t0;j; f0;j; τj;A+;j; φ0;+;j); (5) j=1 N X H×(t) = Ψ(t; t0;j; f0;j; τj;A×;j; φ0;×;j); (6) j=1 so that the two polarizations have independent amplitude and initial phase, but t0, f0, and τ are common parameters.